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Differentiation question

  1. Jul 12, 2015 #1
    1. The problem statement, all variables and given/known data
    A man wishes to fence in a rectangular enclosure of area 128m^2.One side of the enclosure is formed by part of a brick wall already in position.
    What is the least possible length of fencing required for the other three sides?

    2. Relevant equations



    3. The attempt at a solution
    I called one side x and one side y.

    So far I have the relationship

    xy = 128
    y = 128/x

    I was going to differentiate that but that won't help me since equating the result to 0 give me nothing.

    I was thinking the perimeter might help me but.. I don't know what to do with it. Help is greatly appreciated
     
  2. jcsd
  3. Jul 12, 2015 #2

    Orodruin

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    Your problem is to minimise the length of the fence. A good place to start is to find an expression for the length of the fence.
     
  4. Jul 12, 2015 #3
    Okay i'll call the length L , so L = 2x + y?
     
  5. Jul 12, 2015 #4

    Orodruin

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    Correct so far. Now what is the minimal L which encloses 128 m2?
     
  6. Jul 12, 2015 #5
    okay so I did this L = 2x + y .
    L = 2x + (128/x) [from my first area expression]
    L = (2x^2 + 128)/x
    dL/dx = -(2x^2 + 128)/x^2 + 4

    For max/min dL/dx = 0
    2x^2 + 128 = 4x^2
    x = +/- 8
    For L to be a minimum x= 8.
    So Lmin = 16 + 16 = 32m
     
  7. Jul 12, 2015 #6

    Orodruin

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    Yes, there is also no way you can make a side with length -8 m. Be mindful of the domains you have to consider.

    It is also easier to just differentiate 2x + 128/x directly than rewriting it as (2x^2 + 128)/x. The result will of course be the same.
     
  8. Jul 12, 2015 #7
    Oh yeah true, haha wasn't thinking. I truly appreciate it!
     
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